Area Objects and Spatial Autocorrelation Chapter 7 Geographic Information Analysis O’Sullivan and Unwin

Types of Area Objects: Natural Areas Boundaries defined by natural phenomena –Lake, forest, rock outcrop Self-defining Subjective mapping by surveyor –Open to uncertainty Fussiness of boundaries Small unmapped inclusions E.g. soil maps

Types of Area Objects: Fiat or Command Regions Boundaries imposed by humans –Countries, states, census tracts Can be misleading sample of underlying social reality –Boundaries don’t relate to underlying patterns –Boundaries arbitrary or modifiable –Analyses often artifacts of chosen boundaries (MAUP) –Relationships on macrolevel not always same as microlevel

Types of Area Objects: Raster Areas Space divided into raster grid Area objects are uniform and identical and tessellate the region Data structures on squares, hexagons, or triangular mesh

Relationships of Areas Isolated Overlapping Completely contained within each other Planar enforced –Mesh together neatly and completely cover study region –Fundamental assumption of many GIS data models

Storing Area Objects Complete polygons –Doesn’t work for planar enforced areas Store boundary segments –Link boundary segments to build areas –Difficult to transfer data between systems

Geometric Properties of Areas: Area Superficially obvious, but difficult in practice Uses coordinates of vertices to find areas of multiple trapezoids Raster coded data –Count pixels and multiply

Geometric Properties of Areas: Skeleton Internal network of lines –Each point is equidistant nearest 2 edges of boundary Single central point is farthest from boundary –Representative point object location f area object

Geometric Properties of Areas: Shape Set of relationships of relative position between point on their perimeters, unaffected by change in scale Difficult to quantify, can relate to known shape –Compactness ratio =  a/a 2 –Elongation Ratio = L 1 /L 2 –Form Ratio = a/L 1 2 –Radial Line Index

Geometric Properties of Areas: Spatial Pattern & Fragmentation Spatial Pattern Patterns of multiple areas Evaluated by contact numbers –No. of areas that share a common boundary with each area Fragmentation Extent to which the spatila pattern is broken up. –Used commonly in ecology

Spatial Autocorrelation: Review Data from near locations more likely to be similar than data from distant locations Any set of spatial data likely to have characteristic distance at which it is correlated with itself Samples from spatial data are not truly random.

Runs on Serial Data One-Dimensional Autocorrelation Is a series likely to have occurred randomly? Counts runs of same data and compares Z-scores using calculated expected values Nonfree sampling –Probabilities change based on previous trials (e.g. dealing cards) –Most common in GIS data Free sampling –Probability constant (e.g. flipping coin) –Math much easier, so used to estimate nonfree sampling

Joins Count Two-Dimensional Autocorrelation Is a spatial pattern likely to have occurred randomly? Count number of possible joins between neighbors –Rook’s Case = N-S-E-W neighbors –Queen’s Case = Adds diagonal neighbors Compares Z-scores using expected values from free sampling probabilities Only works for binary data

Joins Count Statistic Real World Uses? Was the spatial pattern of 2000 Bush-Gore electoral outcomes random? Build an adjacency matrix (49 x 49) Join TypeZ-Score Bush-Bush Gore-Gore Bush-Gore

Other Measures of Spatial Autocorrelation Moran’s I Translates nonspatial correlation measures to spatial context Applied to numerical ratio or interval data Evaluates summed covariances corrected for sample size I < 0, Negative Autocorrelation I > 0, Positive Autocorrelation Σ(y i -y) 2 - n I = ΣΣw ij ΣΣw ij (y i -y)(y j -y) --

Other Measures of Spatial Autocorrelation Geary’s Contiguity Ratio C Similar to Moran’s I C = 1, No auto correlation 0 < C < 1, Positive autocorrelation C > 1 Negative autocorrelation Σ(y i -y) 2 - n-1 C = 2ΣΣw ij ΣΣw ij (y i -y j ) 2

Other Measures of Spatial Autocorrelation Weighted Matrices Weights can be added to calculations of Moran’s I or Geary’s C –e.g. weight state boundaries based on length of borders Lagged autocorrelation weights in the matrix in which nonadjacent spatial autocorrelation is tested for. –e.g. CA and UT are neighbors at a lag of 2

Local Indicators of Spatial Association (LISA) Where are the data patterns within the study region? Disaggregate measures of autocorrelation Describe extent to which particular areal units are similar to their neighbors Nonstationarity of data –When clusters of similar values found in specific sub- regions of study Tests: G, I, &C